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EE133 – Winter 2002 Cookbook Filter Guide Welcome to the Cookbook Filter Guide!

Don’t have enough time to spice out that perfect filter before Aunt Thelma comes down for dinner? Well this handout is for you! The following pages detail a fast set of steps towards the design and creation of passive filters for practical use in communications and . Amaze your friends and neighbors as you do twice the amount of work in half the time!

Here’s a brief overview of the steps:

® Specify your filter type ® Implement a low-pass version of your filter ® Transform it to what you really want (high-pass, bandpass, bandstop) ® Simulate and Iterate

® Step 1: What Filter Do I Want?

This is where you have to do most of your thinking. It’s no good to cook up a nice steak dinner if Auntie Thelma turned vegetarian last year. Here are the specifications that are most often quoted with filters:

Filter Type (Low-Pass, High-Pass, Band-Pass, Band-Stop) Center (rad/s or Hz) Bandwidth (rad/s or Hz) Cut-off/Roll-off rate (dB) Minimum Attenuation Required in Stopband Input Impedance (ohms) Output Impedance (ohms) Overshoot in Step Response in Step Response

We will be creating filters through a method known as the insertion loss method. Insertion Loss, or the Power Loss Ratio PLR.is defined as: PowerSource 1 PLR = = 2 PowerDelivered 1- G(w) G is our famliar reflection coefficient. It turns out that this is expressible in the following form of : M(w2 ) P =1 + LR N(w 2) Where M and N are two real polynomials. Which simply means that we can define an arbitrary filter response and use this formula to match it to real components, thus allowing us to make it physically realizable. There are several standard filter responses, each with their own advanatages and drawbacks

Standard Filter Responses · Butterworth (AKA Maximally Flat or Binomial) Filters Butterworth filters are general purpose filters. Another common name for them is a maximally flat filter, which refers to the relatively flat magnitude response in the pass- band. Attenuation is –3dB at the design with a –20dB/decade roll off per pole above the cutoff frequency. 1 EE133 – Winter 2002 Cookbook Filter Guide

· Chebyshev (AKA equal magnitude) Filter Chebyshev filters have a steeper attenuation above the cutoff frequency, but at the expense of amplitude “ripples” in the pass-band. For a given number of poles, a steeper cutoff can be achieved by allowing more pass-band ripple. The cutoff frequency is defined at the point at which the response falls below the ripple band of the pass-band.

· Bessel (AKA Linear Phase or Maximally Flat Time Delay) Filters Sometimes a design requires a filter to have a linear phase in order to avoid signal distortion. In general a good phase output (i.e. linear) always comes at the expense of a good magnitude response (i.e. fast attenuation).

· Elliptic Filters For the previous filters, as the frequency gets progressively further from the center frequency the attenuation increases. Sometimes a design only requires a minimum attenuation in the stop-band. This relaxes some constraints on the response which allow a better cutoff rate. However, this filter has ripples in both the and stop-band.

® Step 2: Prototyping a Low-Pass Design Having finally specified your filter, it’s time to prototype a low-pass version of your filter. Although it seems counterintuitive to spend time on a filter that doesn’t even necessarily have the pass- band/stop-band characteristics of your desired filter, it will become apparent that there is relation between the values derived for a low-pass situation and the other filter types.

As an example we will design a low-pass filter for a source impedance of 50 ohm, a cut-off frequency of 1MHz and which requires a minimum attenuation of 40dB at 10MHz.

Determine the type of filter and N, the order of the filter: First we have to determine which of the filter types we want to use. Do we care about having linear phase? Or is maxmium cut-off attenuation the critical factor? Once that’s been done, we can then determine the order of the filter necessary to fit the required attenuation spec. Usually, we refer to a graph like the one below showing the attenuation characteristics for various N versus normalized frequency for a particular filter type.

In this case, we decide that a flat magnitude response in the most appropriate. So then we look at the graph.The definition of normalized frequency is: w fnorm = -1 wc we find that wc=1MHz wo=10MHz which means the normalized frequency is 9. Looking on the graph we see that N=2 will easily satisfy our requirements for 40dB of attenuation.

2 EE133 – Winter 2002 Cookbook Filter Guide

Design a Normalized Low-Pass Filter using a Table: Once that is done, we can now design a second order for a source impedance of 1 ohm, a cut-off frequency of 1 rad/sec. As shown in the figure below, we use one of two equivalent ladder circuits. Note the way the element values are numbered, with g0 at the generator to gN+1 at the load.

How to read this chart:

go= generator resistance or a generator conductance

gk= for series or a capacitance for shunt

gN+1= load resistance if gN is a shunt C or a load conductance if gN is a series L

A key point is that the components alternate between shunt and series. Note that during out prototyping, inductors are always series, capacitors are always shunt. The only difference is whether or not the first element is series or shunt.

3 EE133 – Winter 2002 Cookbook Filter Guide

To design a filter of a particular response (i.e. Butterworth, or Bessel) there is a unique ratio of components to be used. These ratios are usually kept is handy tables like the one below.

Element Values for Butterworth (Maximally Flat) Low-Pass Filter Prototypes (go=1,wc=1,N=1 to 10)

N g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 1 2.0000 1.0000 2 1.4142 1.4142 1.0000 3 1.0000 2.0000 1.0000 1.0000 4 0.7654 1.8478 1.8478 0.7654 1.0000 5 0.6180 1.6180 2.0000 1.6180 0.6180 1.0000 6 0.5176 1.4142 1.9318 1.9318 1.4142 0.5176 1.0000 7 0.4450 1.2470 1.8019 2.0000 1.8019 1.2470 0.4450 1.0000 8 0.3902 1.1111 1.6629 1.9615 1.9615 1.6629 1.1111 0.3902 1.0000 9 0.3473 1.0000 1.5321 1.8794 2.0000 1.8794 1.5321 1.0000 0.3473 1.0000 10 0.3129 0.9080 1.4142 1.7820 1.9754 1.9754 1.7820 1.4142 0.9080 0.3129 1.0000

For a second order, we see that g1 and g2 must both equal 1.4142 and g3, the load, must equal the go, the source resistance. So the following circuit will get us a low pass filter centered about 1 rad/sec.

® Step 3: Transforming the Low-Pass Prototype to the Desired Filter

All we have to do now is transform the prototype into something we can really use. The tables assume a set impedance Rs=Rl=1 ohm (except for Chebyshev filters, where the source and load resistance differ by a set amount). In addition, the filter’s cutoff frequency is 1 rad/sec, a fairly useless value in general. So we need to scale for impedance and frequency.

Let’s start with impedance scaling. To take this into account, we need to multiply the impedances of the prototype by the desired Rs. Since the prototype input impedance is always 1 ohm, we can simply multiply by the ratio. Rl naturally needs to scale proportionally, while all capacitors need to be divided by Rs and all inductors need to be multiplied by Rs.

Now that we have these original values, we need to swap components in order to obtain the correct filter type. Below are the substitutions.

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w1 - w2 D = is your friend w0 D is a term to describe the sharpness of the filter. If you look at it really hard, you’ll see it’s actually the inverse of Q. In the center frequency is not chosen as the arithmetic mean, but the geometric mean. In other words:

w1 + w2 wo ¹ 2

wo = w1w2

This allows for the math to work out such that straight substitutions are allowable. What this means is that often times, you’ll pick your bandwidth and just go for a center frequency in that area. Of course, you could work out the math to get exactly the center frequency you want, but sometimes, you really don’t want to think that hard about a minor filter.

With that all said here are the transformations. The formulas on the next page were derived by comparing the frequency dependent impedances and mapping the cutoff frequency to different situations. In each one, the Lk or Ck from your prototype is coverted into a new Lk’ or Ck’. A derivation of these formulas can by found in Posner’s Microwave Transmission. NOTE: These formulas ALREADY take into account the scaling for impedances.

5 EE133 – Winter 2002 Cookbook Filter Guide TRANSFORMATION TABLE

Low-Pass

1 L' = R L k w s k c Simple impedance and frequency scaling. 1 C C' = k k w R c s High-Pass

1 1 C' = k w R L c s k Note the swapping of capacitors and inductors: 1 R L' = s k w C c k Bandpass

' 1 RsLk LLk = wo D Series Lk transforms to a series LC circuit defined by: ' 1 D CLk = wo RsLk

' 1 RsD LCk = wo Ck Shunt Ck transforms to a parallel LC circuit with element values: ' 1 Ck CCk = wo RsD Bandstop

' 1 LLk = DRsLk wo Series inductor Lk transforms to a parallel LC circuit defined by: ' 1 1 CLk = wo DRsLk

' 1 Rs LCk = wo DCk Shunt capacitor Ck transforms to a series LC circuit with element values: ' 1 DCk CCk = wo Rs

In our case, we can see that we need to multiply our L value by Rs/wc and our C value by 1/wcRs. L=11.25uH and C=4.50nF. Now we test this in HSPICE or Matlab and see if our response is good enough for our purposes. If not, we iterate, making educated tweaks. Note that we’ve mainly designed for the frequency domain in this methodology, which is fairly straightforward. Time-domain characterstics, such as ringing, or overshoot, are often equally important but the methods for dealing with them are completely different. But that’s for another handout to cover…

6 EE133 – Winter 2002 Cookbook Filter Guide Element Values for Butterworth (Maximally Flat) Low-Pass Filter Prototypes (go=1,wc=1,N=1 to 10) N g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 1 2.0000 1.0000 2 1.4142 1.4142 1.0000 3 1.0000 2.0000 1.0000 1.0000 4 0.7654 1.8478 1.8478 0.7654 1.0000 5 0.6180 1.6180 2.0000 1.6180 0.6180 1.0000 6 0.5176 1.4142 1.9318 1.9318 1.4142 0.5176 1.0000 7 0.4450 1.2470 1.8019 2.0000 1.8019 1.2470 0.4450 1.0000 8 0.3902 1.1111 1.6629 1.9615 1.9615 1.6629 1.1111 0.3902 1.0000 9 0.3473 1.0000 1.5321 1.8794 2.0000 1.8794 1.5321 1.0000 0.3473 1.0000 10 0.3129 0.9080 1.4142 1.7820 1.9754 1.9754 1.7820 1.4142 0.9080 0.3129 1.0000

Element Values for Bessel (Maximally Flat Time Delay) Low-Pass Filter Prototypes (go=1,wc=1,N=1 to 10) N g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 1 2.0000 1.0000 2 1.5774 0.4226 1.0000 3 1.2550 0.5528 0.1922 1.0000 4 1.0598 0.5116 0.3181 0.1104 1.0000 5 0.9303 0.4577 0.3312 0.2090 0.0718 1.0000 6 0.8377 0.4116 0.3158 0.2364 0.1480 0.0505 1.0000 7 0.7677 0.3744 0.2944 0.2378 0.1778 0.1104 0.0375 1.0000 8 0.7125 0.3446 0.2735 0.2297 0.1867 0.1387 0.0855 0.0289 1.0000 9 0.6678 0.3203 0.2547 0.2184 0.1859 0.1506 0.1111 0.0682 0.0230 1.0000 10 0.6305 0.3002 0.2384 0.2066 0.1808 0.1539 0.1240 0.0911 0.0557 0.0187 1.0000

Element Values for Chebyshev (Equal-Ripple) Low-Pass Filter Prototypes (go=1,wc=1,N=1 to 10) 0.5dB Ripple Band

N g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 1 0.6986 1.0000 2 1.4029 0.7071 1.9841 3 1.5963 1.0967 1.5963 1.0000 4 1.6703 1.1926 2.3661 0.8419 1.9841 5 1.7058 1.2296 2.5408 1.2296 1.7058 1.0000 6 1.7254 1.2479 2.6064 1.3137 2.4758 0.8696 1.9841 7 1.7372 1.2583 2.6381 1.3444 2.6381 1.2583 1.7372 1.0000 8 1.7451 1.2647 2.6564 1.3590 2.6964 1.3389 2.5093 0.8796 1.9841 9 1.7504 1.2690 2.6678 1.3673 2.7239 1.3673 2.6678 1.2690 1.7504 1.0000 10 1.7543 1.2721 2.6754 1.3725 2.7392 1.3806 2.7231 1.3485 2.5239 0.8842 1.9841

3.0dB Ripple Band

N g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 1 1.9953 1.0000 2 3.1013 0.5339 5.8095 3 3.3487 0.7117 3.3487 1.0000 4 3.4289 0.7483 4.3471 0.5920 5.8095 5 3.4817 0.7618 4.5381 0.7618 3.4817 1.0000 6 3.5045 0.7685 4.6061 0.7929 4.4641 0.6033 5.8095 7 3.5182 0.7723 4.6386 0.8039 4.6386 0.7723 3.5182 1.0000 8 3.5277 0.7745 4.6575 0.8089 4.6990 0.8018 4.4990 0.6073 5.8095 9 3.5340 0.7760 4.6692 0.8118 4.7272 0.8118 4.6692 0.7760 3.5340 1.0000 10 3.5384 0.7771 4.6768 0.8136 4.7425 0.8164 4.7260 0.8051 4.5142 0.6091 5.8095

Please note that these tables assume the following: 2 w = c LC Which means that your filters will have a smaller bandwidth than you usually expect if you define 3dB as your cutoff frequency. You may easy compensate this by just roughly doubling wc

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